| L(s) = 1 | − 1.70·2-s + 3-s + 0.911·4-s − 1.70·6-s − 3.94·7-s + 1.85·8-s + 9-s − 5.90·11-s + 0.911·12-s + 3.29·13-s + 6.72·14-s − 4.99·16-s + 2.70·17-s − 1.70·18-s − 2.35·19-s − 3.94·21-s + 10.0·22-s − 0.584·23-s + 1.85·24-s − 5.61·26-s + 27-s − 3.59·28-s − 3.91·29-s + 2.70·31-s + 4.80·32-s − 5.90·33-s − 4.61·34-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.577·3-s + 0.455·4-s − 0.696·6-s − 1.49·7-s + 0.656·8-s + 0.333·9-s − 1.78·11-s + 0.263·12-s + 0.912·13-s + 1.79·14-s − 1.24·16-s + 0.656·17-s − 0.402·18-s − 0.540·19-s − 0.860·21-s + 2.14·22-s − 0.121·23-s + 0.379·24-s − 1.10·26-s + 0.192·27-s − 0.679·28-s − 0.726·29-s + 0.486·31-s + 0.849·32-s − 1.02·33-s − 0.791·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6558065109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6558065109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 0.584T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 0.0208T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 2.81T + 53T^{2} \) |
| 59 | \( 1 + 4.69T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 0.781T + 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 + 2.45T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.292928525179429893909597036873, −8.454684471049539044758191190090, −7.911229861817684167447881406575, −7.18348276896578874171606924812, −6.27249498231025184251109000148, −5.31313643518395762088516634701, −4.03607478058044305324747557416, −3.10169617875115115336089245772, −2.17189411661529837186689437601, −0.61628103870051830695030692803,
0.61628103870051830695030692803, 2.17189411661529837186689437601, 3.10169617875115115336089245772, 4.03607478058044305324747557416, 5.31313643518395762088516634701, 6.27249498231025184251109000148, 7.18348276896578874171606924812, 7.911229861817684167447881406575, 8.454684471049539044758191190090, 9.292928525179429893909597036873
